Question

Figure ABCD is parallelogram with point A (-3,2) what rule would rotate the figure 90 counterclockwise

Answer 1

Answer:

(x, y) ⇒ (-y, x)

Explanation:

A point at y=1 on the y-axis will rotate to the point x = -1 on the x-axis when it is rotated 90° CCW about the origin. Hence the value of x for the image point is the opposite of the y-value of the original point.

A point at x=1 on the x-axis will rotate to the point y=1 on the y-axis when it is rotated 90° CCW about the origin. Hence the value of y for the image point is the x-value of the original point.

In summary, ...

... (x, y) ⇒ (-y, x)

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Comment on rotation matrices

The rotation matrix for rotatation through the angle θ CCW about the origin is ...

$\left[\begin{array}{cc}\cos{(\theta)}&-\sin{(\theta)}\\\sin{(\theta)}&\cos{(\theta)}\end{array}\right]$

When θ = 90°, this matrix becomes ...

$\left[\begin{array}{cc}0&-1\\1&0\end{array}\right]$

and multiplication by coordinates (x, y) gives ...

$\left[\begin{array}{cc}0&-1\\1&0\end{array}\right]\times\left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}-y\\x\end{array}\right]$